The spectrogram of a signal throws away the phase information, but it is said to be possible to reconstruct the signal only from the spectrogram [1] via
$$s(t) = \frac{1}{2\pi s^*(0)}\int_{-\infty}^{\infty}\frac{M_{SP}(\theta,t)}{A_h(-\theta,t)}e^{-j\theta t/2}d\theta$$ where $M_{SP}$ is the chariteristic function of spectrogram and ${A_h(\theta,t)}$ is the ambiguity function of the window.

The formula seems to work in theory. Could anybody explain why it works? Some phase information is thrown away after all. Also, is it a good reconstruction tool used in practice? It seems to me that nobody uses this as the reconstruction tool.